MikeH

Answer: He lets Jack go and decides to give him a one day head start before pursuing him again out of affection for Elizabeth, whom along with Will wanted Jack to be spared from execution for his good deeds.

Phaneron

Answer: At no point did he actually "let" him escape on purpose. Jack was always just too witty for Norrington (who is kinda slow). That's counting 3 occasions where Norrington failed to capture Jack, but perhaps you can be more specific as to which escape you are referring to.

lionhead

The escape is at the end of the movie. When Elizabeth and Will help Jack escape. Norrigton is asked if they are going to go after Jack but Norrigton decides to give him a head start.

In that case then, Norrington is confident that the British Navy can capture 1 pirate with relative ease.

Ssiscool

Answer: He is overly-confident that he is more capable than Jack and will easily catch him.

raywest

Answer: In addition to the other reasons mentioned, Governor Swann suggests it, and not wanting to cross him may have contributed to Norrington's decision.